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The Fundamental Theorem of Arithmetic states:

* Apart from the rearrangement of factors, _ a positive whole number can be expressed as _ a product of

Euclid essentially produced a proof but the first full proof was made by Gauss.

Proving that every number can be expressed as the product of prime numbers is fairly

simple, it is the *_uniqueness_* of the representation that is a challenge.

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To show why this is not as easy as you might think at first glance, let's look at the ring of numbers that are all multiples of 6.

Firstly, we have the concept of prime numbers, because, for example, 72 is not prime because it can be written as 6*12.

The number 12, however, cannot be written as the product of two other numbers, so it must be, in some sense, in this

system, considered to be prime. Similarly 18 is "prime" in this system.

Now think about 216. We can write that as 12*18, as you can check, but we can also write it as 6*6*6. We have written

216 as the product of "primes" in two different ways.

Something has gone wrong.

Now, obviously, the something that's gone wrong is that these aren't really "primes" in the sense we usually mean,

but in the complex numbers the usual primes aren't always primes either. For example, EQN:5=(2+i)(2-i).

Some care required.

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As a more compelling example, consider the ring of integers EQN:Z extended by EQN:\sqrt{-5} giving EQN:Z[\sqrt{-5}].

In this ring EQN:6=2.3=(1+\sqrt{-5})(1-\sqrt{-5}) showing that 6 has more than one factorisation.

A simple arguments about modulus shows that 2, 3, EQN:1+\sqrt{-5} and EQN:1-\sqrt{-5} are all prime numbers in the usual sense,

and this shows that in the ring EQN:Z[\sqrt{-5}] the unique factorisation of integers is not true.